LMFDB - Embedded newform 4800.2.a.cv.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4800,2,Mod(1,4800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4800 = 2^{6} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4800.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$38.3281929702$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 480)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4800.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.00000 q^{7} +1.00000 q^{9} +4.47214 q^{11} -4.47214 q^{13} -4.47214 q^{17} -2.00000 q^{21} -4.00000 q^{23} +1.00000 q^{27} -4.00000 q^{29} +8.94427 q^{31} +4.47214 q^{33} +4.47214 q^{37} -4.47214 q^{39} +10.0000 q^{41} -4.00000 q^{43} -8.00000 q^{47} -3.00000 q^{49} -4.47214 q^{51} +4.47214 q^{53} -13.4164 q^{59} -10.0000 q^{61} -2.00000 q^{63} -8.00000 q^{67} -4.00000 q^{69} -8.94427 q^{71} -8.94427 q^{73} -8.94427 q^{77} -8.94427 q^{79} +1.00000 q^{81} -4.00000 q^{83} -4.00000 q^{87} +6.00000 q^{89} +8.94427 q^{91} +8.94427 q^{93} +17.8885 q^{97} +4.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{7} + 2 q^{9} - 4 q^{21} - 8 q^{23} + 2 q^{27} - 8 q^{29} + 20 q^{41} - 8 q^{43} - 16 q^{47} - 6 q^{49} - 20 q^{61} - 4 q^{63} - 16 q^{67} - 8 q^{69} + 2 q^{81} - 8 q^{83} - 8 q^{87} + 12 q^{89}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^7 + 2 * q^9 - 4 * q^21 - 8 * q^23 + 2 * q^27 - 8 * q^29 + 20 * q^41 - 8 * q^43 - 16 * q^47 - 6 * q^49 - 20 * q^61 - 4 * q^63 - 16 * q^67 - 8 * q^69 + 2 * q^81 - 8 * q^83 - 8 * q^87 + 12 * q^89

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	8.94427	1.60644	0.803219	−	0.595683i	$-0.203119\pi$
$31$	8.94427	1.60644	0.803219	+	0.595683i	$0.203119\pi$
$32$	0	0
$33$	4.47214	0.778499
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	0	0
$39$	−4.47214	−0.716115
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−4.47214	−0.626224
$52$	0	0
$53$	4.47214	0.614295	0.307148	−	0.951662i	$-0.400625\pi$
$53$	4.47214	0.614295	0.307148	+	0.951662i	$0.400625\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.4164	−1.74667	−0.873334	−	0.487122i	$-0.838047\pi$
$59$	−13.4164	−1.74667	−0.873334	+	0.487122i	$0.838047\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−8.94427	−1.06149	−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427	−1.06149	−0.530745	+	0.847532i	$0.678088\pi$
$72$	0	0
$73$	−8.94427	−1.04685	−0.523424	−	0.852072i	$-0.675346\pi$
$73$	−8.94427	−1.04685	−0.523424	+	0.852072i	$0.675346\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.94427	−1.01929
$78$	0	0
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.00000	−0.428845
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	8.94427	0.937614
$92$	0	0
$93$	8.94427	0.927478
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17.8885	1.81631	0.908153	−	0.418638i	$-0.137492\pi$
$97$	17.8885	1.81631	0.908153	+	0.418638i	$0.137492\pi$
$98$	0	0
$99$	4.47214	0.449467
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	4.47214	0.424476
$112$	0	0
$113$	4.47214	0.420703	0.210352	−	0.977626i	$-0.432539\pi$
$113$	4.47214	0.420703	0.210352	+	0.977626i	$0.432539\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.47214	−0.413449
$118$	0	0
$119$	8.94427	0.819920
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	10.0000	0.901670
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	−4.47214	−0.390732	−0.195366	−	0.980730i	$-0.562590\pi$
$131$	−4.47214	−0.390732	−0.195366	+	0.980730i	$0.562590\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.47214	0.382080	0.191040	−	0.981582i	$-0.438814\pi$
$137$	4.47214	0.382080	0.191040	+	0.981582i	$0.438814\pi$
$138$	0	0
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−20.0000	−1.67248
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−8.94427	−0.727875	−0.363937	−	0.931423i	$-0.618568\pi$
$151$	−8.94427	−0.727875	−0.363937	+	0.931423i	$0.618568\pi$
$152$	0	0
$153$	−4.47214	−0.361551
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.4164	1.07075	0.535373	−	0.844616i	$-0.320171\pi$
$157$	13.4164	1.07075	0.535373	+	0.844616i	$0.320171\pi$
$158$	0	0
$159$	4.47214	0.354663
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.47214	−0.340010	−0.170005	−	0.985443i	$-0.554378\pi$
$173$	−4.47214	−0.340010	−0.170005	+	0.985443i	$0.554378\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−13.4164	−1.00844
$178$	0	0
$179$	22.3607	1.67132	0.835658	−	0.549250i	$-0.185087\pi$
$179$	22.3607	1.67132	0.835658	+	0.549250i	$0.185087\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−20.0000	−1.46254
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	8.94427	0.647185	0.323592	−	0.946197i	$-0.395109\pi$
$191$	8.94427	0.647185	0.323592	+	0.946197i	$0.395109\pi$
$192$	0	0
$193$	−8.94427	−0.643823	−0.321911	−	0.946770i	$-0.604325\pi$
$193$	−8.94427	−0.643823	−0.321911	+	0.946770i	$0.604325\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.47214	0.318626	0.159313	−	0.987228i	$-0.449072\pi$
$197$	4.47214	0.318626	0.159313	+	0.987228i	$0.449072\pi$
$198$	0	0
$199$	26.8328	1.90213	0.951064	−	0.308994i	$-0.0999924\pi$
$199$	26.8328	1.90213	0.951064	+	0.308994i	$0.0999924\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	8.00000	0.561490
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−26.8328	−1.84725	−0.923624	−	0.383301i	$-0.874787\pi$
$211$	−26.8328	−1.84725	−0.923624	+	0.383301i	$0.874787\pi$
$212$	0	0
$213$	−8.94427	−0.612851
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−17.8885	−1.21435
$218$	0	0
$219$	−8.94427	−0.604398
$220$	0	0
$221$	20.0000	1.34535
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	−8.94427	−0.588490
$232$	0	0
$233$	13.4164	0.878938	0.439469	−	0.898258i	$-0.355167\pi$
$233$	13.4164	0.878938	0.439469	+	0.898258i	$0.355167\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.94427	−0.580993
$238$	0	0
$239$	−8.94427	−0.578557	−0.289278	−	0.957245i	$-0.593415\pi$
$239$	−8.94427	−0.578557	−0.289278	+	0.957245i	$0.593415\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−4.47214	−0.282279	−0.141139	−	0.989990i	$-0.545077\pi$
$251$	−4.47214	−0.282279	−0.141139	+	0.989990i	$0.545077\pi$
$252$	0	0
$253$	−17.8885	−1.12464
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.4164	−0.836893	−0.418446	−	0.908242i	$-0.637425\pi$
$257$	−13.4164	−0.836893	−0.418446	+	0.908242i	$0.637425\pi$
$258$	0	0
$259$	−8.94427	−0.555770
$260$	0	0
$261$	−4.00000	−0.247594
$262$	0	0
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000	0.367194
$268$	0	0
$269$	20.0000	1.21942	0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000	1.21942	0.609711	+	0.792624i	$0.291286\pi$
$270$	0	0
$271$	−26.8328	−1.62998	−0.814989	−	0.579477i	$-0.803257\pi$
$271$	−26.8328	−1.62998	−0.814989	+	0.579477i	$0.803257\pi$
$272$	0	0
$273$	8.94427	0.541332
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−13.4164	−0.806114	−0.403057	−	0.915175i	$-0.632052\pi$
$277$	−13.4164	−0.806114	−0.403057	+	0.915175i	$0.632052\pi$
$278$	0	0
$279$	8.94427	0.535480
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−20.0000	−1.18056
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	17.8885	1.04865
$292$	0	0
$293$	4.47214	0.261265	0.130632	−	0.991431i	$-0.458299\pi$
$293$	4.47214	0.261265	0.130632	+	0.991431i	$0.458299\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.47214	0.259500
$298$	0	0
$299$	17.8885	1.03452
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	−12.0000	−0.689382
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	14.0000	0.796432
$310$	0	0
$311$	8.94427	0.507183	0.253592	−	0.967311i	$-0.418388\pi$
$311$	8.94427	0.507183	0.253592	+	0.967311i	$0.418388\pi$
$312$	0	0
$313$	26.8328	1.51668	0.758340	−	0.651859i	$-0.226011\pi$
$313$	26.8328	1.51668	0.758340	+	0.651859i	$0.226011\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	13.4164	0.753541	0.376770	−	0.926307i	$-0.377035\pi$
$317$	13.4164	0.753541	0.376770	+	0.926307i	$0.377035\pi$
$318$	0	0
$319$	−17.8885	−1.00157
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−10.0000	−0.553001
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	4.47214	0.245072
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.94427	0.487226	0.243613	−	0.969873i	$-0.421667\pi$
$337$	8.94427	0.487226	0.243613	+	0.969873i	$0.421667\pi$
$338$	0	0
$339$	4.47214	0.242893
$340$	0	0
$341$	40.0000	2.16612
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	−4.47214	−0.238705
$352$	0	0
$353$	−22.3607	−1.19014	−0.595069	−	0.803674i	$-0.702875\pi$
$353$	−22.3607	−1.19014	−0.595069	+	0.803674i	$0.702875\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	8.94427	0.473381
$358$	0	0
$359$	−17.8885	−0.944121	−0.472061	−	0.881566i	$-0.656490\pi$
$359$	−17.8885	−0.944121	−0.472061	+	0.881566i	$0.656490\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	9.00000	0.472377
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	−8.94427	−0.464363
$372$	0	0
$373$	13.4164	0.694675	0.347338	−	0.937740i	$-0.387086\pi$
$373$	13.4164	0.694675	0.347338	+	0.937740i	$0.387086\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17.8885	0.921307
$378$	0	0
$379$	−26.8328	−1.37831	−0.689155	−	0.724614i	$-0.742018\pi$
$379$	−26.8328	−1.37831	−0.689155	+	0.724614i	$0.742018\pi$
$380$	0	0
$381$	−2.00000	−0.102463
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	17.8885	0.904663
$392$	0	0
$393$	−4.47214	−0.225589
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.47214	0.224450	0.112225	−	0.993683i	$-0.464202\pi$
$397$	4.47214	0.224450	0.112225	+	0.993683i	$0.464202\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	0	0
$403$	−40.0000	−1.99254
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.0000	0.991363
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	4.47214	0.220594
$412$	0	0
$413$	26.8328	1.32036
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.94427	0.438003
$418$	0	0
$419$	−22.3607	−1.09239	−0.546195	−	0.837658i	$-0.683924\pi$
$419$	−22.3607	−1.09239	−0.546195	+	0.837658i	$0.683924\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	0	0
$426$	0	0
$427$	20.0000	0.967868
$428$	0	0
$429$	−20.0000	−0.965609
$430$	0	0
$431$	17.8885	0.861661	0.430830	−	0.902433i	$-0.358221\pi$
$431$	17.8885	0.861661	0.430830	+	0.902433i	$0.358221\pi$
$432$	0	0
$433$	−26.8328	−1.28950	−0.644751	−	0.764392i	$-0.723039\pi$
$433$	−26.8328	−1.28950	−0.644751	+	0.764392i	$0.723039\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−8.94427	−0.426887	−0.213443	−	0.976955i	$-0.568468\pi$
$439$	−8.94427	−0.426887	−0.213443	+	0.976955i	$0.568468\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	44.7214	2.10585
$452$	0	0
$453$	−8.94427	−0.420239
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	−4.47214	−0.208741
$460$	0	0
$461$	32.0000	1.49039	0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000	1.49039	0.745194	+	0.666847i	$0.232357\pi$
$462$	0	0
$463$	−6.00000	−0.278844	−0.139422	−	0.990233i	$-0.544524\pi$
$463$	−6.00000	−0.278844	−0.139422	+	0.990233i	$0.544524\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	13.4164	0.618195
$472$	0	0
$473$	−17.8885	−0.822516
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.47214	0.204765
$478$	0	0
$479$	8.94427	0.408674	0.204337	−	0.978901i	$-0.434496\pi$
$479$	8.94427	0.408674	0.204337	+	0.978901i	$0.434496\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	0	0
$483$	8.00000	0.364013
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	−24.0000	−1.08532
$490$	0	0
$491$	−4.47214	−0.201825	−0.100912	−	0.994895i	$-0.532176\pi$
$491$	−4.47214	−0.201825	−0.100912	+	0.994895i	$0.532176\pi$
$492$	0	0
$493$	17.8885	0.805659
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17.8885	0.802411
$498$	0	0
$499$	17.8885	0.800801	0.400401	−	0.916340i	$-0.368871\pi$
$499$	17.8885	0.800801	0.400401	+	0.916340i	$0.368871\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.00000	0.310881
$508$	0	0
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	17.8885	0.791343
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−35.7771	−1.57347
$518$	0	0
$519$	−4.47214	−0.196305
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	24.0000	1.04945	0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000	1.04945	0.524723	+	0.851273i	$0.324169\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−40.0000	−1.74243
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−13.4164	−0.582223
$532$	0	0
$533$	−44.7214	−1.93710
$534$	0	0
$535$	0	0
$536$	0	0
$537$	22.3607	0.964935
$538$	0	0
$539$	−13.4164	−0.577886
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	−18.0000	−0.772454
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	0	0
$552$	0	0
$553$	17.8885	0.760698
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.3050	−1.32643	−0.663217	−	0.748427i	$-0.730809\pi$
$557$	−31.3050	−1.32643	−0.663217	+	0.748427i	$0.730809\pi$
$558$	0	0
$559$	17.8885	0.756605
$560$	0	0
$561$	−20.0000	−0.844401
$562$	0	0
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	8.94427	0.373652
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−17.8885	−0.744710	−0.372355	−	0.928090i	$-0.621450\pi$
$577$	−17.8885	−0.744710	−0.372355	+	0.928090i	$0.621450\pi$
$578$	0	0
$579$	−8.94427	−0.371711
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	20.0000	0.828315
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	4.47214	0.183959
$592$	0	0
$593$	40.2492	1.65284	0.826419	−	0.563056i	$-0.190374\pi$
$593$	40.2492	1.65284	0.826419	+	0.563056i	$0.190374\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	26.8328	1.09819
$598$	0	0
$599$	−17.8885	−0.730906	−0.365453	−	0.930830i	$-0.619086\pi$
$599$	−17.8885	−0.730906	−0.365453	+	0.930830i	$0.619086\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	0	0
$606$	0	0
$607$	42.0000	1.70473	0.852364	−	0.522949i	$-0.175168\pi$
$607$	42.0000	1.70473	0.852364	+	0.522949i	$0.175168\pi$
$608$	0	0
$609$	8.00000	0.324176
$610$	0	0
$611$	35.7771	1.44739
$612$	0	0
$613$	22.3607	0.903139	0.451570	−	0.892236i	$-0.350864\pi$
$613$	22.3607	0.903139	0.451570	+	0.892236i	$0.350864\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.3607	−0.900207	−0.450104	−	0.892976i	$-0.648613\pi$
$617$	−22.3607	−0.900207	−0.450104	+	0.892976i	$0.648613\pi$
$618$	0	0
$619$	26.8328	1.07850	0.539251	−	0.842145i	$-0.318707\pi$
$619$	26.8328	1.07850	0.539251	+	0.842145i	$0.318707\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	−12.0000	−0.480770
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−20.0000	−0.797452
$630$	0	0
$631$	−8.94427	−0.356066	−0.178033	−	0.984025i	$-0.556973\pi$
$631$	−8.94427	−0.356066	−0.178033	+	0.984025i	$0.556973\pi$
$632$	0	0
$633$	−26.8328	−1.06651
$634$	0	0
$635$	0	0
$636$	0	0
$637$	13.4164	0.531577
$638$	0	0
$639$	−8.94427	−0.353830
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	−16.0000	−0.630978	−0.315489	−	0.948929i	$-0.602169\pi$
$643$	−16.0000	−0.630978	−0.315489	+	0.948929i	$0.602169\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	0	0
$649$	−60.0000	−2.35521
$650$	0	0
$651$	−17.8885	−0.701107
$652$	0	0
$653$	−22.3607	−0.875041	−0.437521	−	0.899208i	$-0.644143\pi$
$653$	−22.3607	−0.875041	−0.437521	+	0.899208i	$0.644143\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.94427	−0.348949
$658$	0	0
$659$	−4.47214	−0.174210	−0.0871048	−	0.996199i	$-0.527762\pi$
$659$	−4.47214	−0.174210	−0.0871048	+	0.996199i	$0.527762\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	20.0000	0.776736
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.0000	0.619522
$668$	0	0
$669$	6.00000	0.231973
$670$	0	0
$671$	−44.7214	−1.72645
$672$	0	0
$673$	26.8328	1.03433	0.517165	−	0.855886i	$-0.326988\pi$
$673$	26.8328	1.03433	0.517165	+	0.855886i	$0.326988\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	31.3050	1.20315	0.601574	−	0.798817i	$-0.294541\pi$
$677$	31.3050	1.20315	0.601574	+	0.798817i	$0.294541\pi$
$678$	0	0
$679$	−35.7771	−1.37300
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−6.00000	−0.228914
$688$	0	0
$689$	−20.0000	−0.761939
$690$	0	0
$691$	−17.8885	−0.680512	−0.340256	−	0.940333i	$-0.610514\pi$
$691$	−17.8885	−0.680512	−0.340256	+	0.940333i	$0.610514\pi$
$692$	0	0
$693$	−8.94427	−0.339765
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−44.7214	−1.69394
$698$	0	0
$699$	13.4164	0.507455
$700$	0	0
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24.0000	0.902613
$708$	0	0
$709$	−14.0000	−0.525781	−0.262891	−	0.964826i	$-0.584676\pi$
$709$	−14.0000	−0.525781	−0.262891	+	0.964826i	$0.584676\pi$
$710$	0	0
$711$	−8.94427	−0.335436
$712$	0	0
$713$	−35.7771	−1.33986
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.94427	−0.334030
$718$	0	0
$719$	35.7771	1.33426	0.667130	−	0.744941i	$-0.267522\pi$
$719$	35.7771	1.33426	0.667130	+	0.744941i	$0.267522\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	0	0
$723$	−10.0000	−0.371904
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.0000	0.667583	0.333792	−	0.942647i	$-0.391672\pi$
$727$	18.0000	0.667583	0.333792	+	0.942647i	$0.391672\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	17.8885	0.661632
$732$	0	0
$733$	−13.4164	−0.495546	−0.247773	−	0.968818i	$-0.579699\pi$
$733$	−13.4164	−0.495546	−0.247773	+	0.968818i	$0.579699\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−35.7771	−1.31787
$738$	0	0
$739$	−35.7771	−1.31608	−0.658041	−	0.752982i	$-0.728615\pi$
$739$	−35.7771	−1.31608	−0.658041	+	0.752982i	$0.728615\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	24.0000	0.876941
$750$	0	0
$751$	26.8328	0.979143	0.489572	−	0.871963i	$-0.337153\pi$
$751$	26.8328	0.979143	0.489572	+	0.871963i	$0.337153\pi$
$752$	0	0
$753$	−4.47214	−0.162974
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−13.4164	−0.487628	−0.243814	−	0.969822i	$-0.578399\pi$
$757$	−13.4164	−0.487628	−0.243814	+	0.969822i	$0.578399\pi$
$758$	0	0
$759$	−17.8885	−0.649313
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	60.0000	2.16647
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	−13.4164	−0.483180
$772$	0	0
$773$	−4.47214	−0.160852	−0.0804258	−	0.996761i	$-0.525628\pi$
$773$	−4.47214	−0.160852	−0.0804258	+	0.996761i	$0.525628\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8.94427	−0.320874
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−40.0000	−1.43131
$782$	0	0
$783$	−4.00000	−0.142948
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	0	0
$789$	4.00000	0.142404
$790$	0	0
$791$	−8.94427	−0.318022
$792$	0	0
$793$	44.7214	1.58810
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.47214	−0.158411	−0.0792056	−	0.996858i	$-0.525238\pi$
$797$	−4.47214	−0.158411	−0.0792056	+	0.996858i	$0.525238\pi$
$798$	0	0
$799$	35.7771	1.26570
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	−40.0000	−1.41157
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20.0000	0.704033
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−26.8328	−0.942228	−0.471114	−	0.882072i	$-0.656148\pi$
$811$	−26.8328	−0.942228	−0.471114	+	0.882072i	$0.656148\pi$
$812$	0	0
$813$	−26.8328	−0.941068
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	8.94427	0.312538
$820$	0	0
$821$	20.0000	0.698005	0.349002	−	0.937122i	$-0.386521\pi$
$821$	20.0000	0.698005	0.349002	+	0.937122i	$0.386521\pi$
$822$	0	0
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	0	0
$831$	−13.4164	−0.465410
$832$	0	0
$833$	13.4164	0.464851
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.94427	0.309159
$838$	0	0
$839$	35.7771	1.23516	0.617581	−	0.786507i	$-0.288113\pi$
$839$	35.7771	1.23516	0.617581	+	0.786507i	$0.288113\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	0	0
$843$	30.0000	1.03325
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−18.0000	−0.618487
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	−17.8885	−0.613211
$852$	0	0
$853$	40.2492	1.37811	0.689054	−	0.724710i	$-0.258026\pi$
$853$	40.2492	1.37811	0.689054	+	0.724710i	$0.258026\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49.1935	1.68042	0.840209	−	0.542263i	$-0.182432\pi$
$857$	49.1935	1.68042	0.840209	+	0.542263i	$0.182432\pi$
$858$	0	0
$859$	44.7214	1.52587	0.762937	−	0.646473i	$-0.223757\pi$
$859$	44.7214	1.52587	0.762937	+	0.646473i	$0.223757\pi$
$860$	0	0
$861$	−20.0000	−0.681598
$862$	0	0
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.00000	0.101885
$868$	0	0
$869$	−40.0000	−1.35691
$870$	0	0
$871$	35.7771	1.21226
$872$	0	0
$873$	17.8885	0.605435
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.4164	0.453040	0.226520	−	0.974007i	$-0.427265\pi$
$877$	13.4164	0.453040	0.226520	+	0.974007i	$0.427265\pi$
$878$	0	0
$879$	4.47214	0.150841
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	0	0
$891$	4.47214	0.149822
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	17.8885	0.597281
$898$	0	0
$899$	−35.7771	−1.19323
$900$	0	0
$901$	−20.0000	−0.666297
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	0	0
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−35.7771	−1.18535	−0.592674	−	0.805443i	$-0.701928\pi$
$911$	−35.7771	−1.18535	−0.592674	+	0.805443i	$0.701928\pi$
$912$	0	0
$913$	−17.8885	−0.592024
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.94427	0.295366
$918$	0	0
$919$	−8.94427	−0.295044	−0.147522	−	0.989059i	$-0.547130\pi$
$919$	−8.94427	−0.295044	−0.147522	+	0.989059i	$0.547130\pi$
$920$	0	0
$921$	−28.0000	−0.922631
$922$	0	0
$923$	40.0000	1.31662
$924$	0	0
$925$	0	0
$926$	0	0
$927$	14.0000	0.459820
$928$	0	0
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	8.94427	0.292822
$934$	0	0
$935$	0	0
$936$	0	0
$937$	53.6656	1.75318	0.876590	−	0.481238i	$-0.159813\pi$
$937$	53.6656	1.75318	0.876590	+	0.481238i	$0.159813\pi$
$938$	0	0
$939$	26.8328	0.875656
$940$	0	0
$941$	32.0000	1.04317	0.521585	−	0.853199i	$-0.325341\pi$
$941$	32.0000	1.04317	0.521585	+	0.853199i	$0.325341\pi$
$942$	0	0
$943$	−40.0000	−1.30258
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.0000	−0.909878	−0.454939	−	0.890523i	$-0.650339\pi$
$947$	−28.0000	−0.909878	−0.454939	+	0.890523i	$0.650339\pi$
$948$	0	0
$949$	40.0000	1.29845
$950$	0	0
$951$	13.4164	0.435057
$952$	0	0
$953$	−31.3050	−1.01407	−0.507033	−	0.861926i	$-0.669258\pi$
$953$	−31.3050	−1.01407	−0.507033	+	0.861926i	$0.669258\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−17.8885	−0.578254
$958$	0	0
$959$	−8.94427	−0.288826
$960$	0	0
$961$	49.0000	1.58065
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	0	0
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−40.2492	−1.29166	−0.645830	−	0.763482i	$-0.723488\pi$
$971$	−40.2492	−1.29166	−0.645830	+	0.763482i	$0.723488\pi$
$972$	0	0
$973$	−17.8885	−0.573480
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.1935	1.57384	0.786920	−	0.617055i	$-0.211675\pi$
$977$	49.1935	1.57384	0.786920	+	0.617055i	$0.211675\pi$
$978$	0	0
$979$	26.8328	0.857581
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16.0000	0.509286
$988$	0	0
$989$	16.0000	0.508770
$990$	0	0
$991$	−44.7214	−1.42062	−0.710310	−	0.703889i	$-0.751445\pi$
$991$	−44.7214	−1.42062	−0.710310	+	0.703889i	$0.751445\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−4.47214	−0.141634	−0.0708170	−	0.997489i	$-0.522561\pi$
$997$	−4.47214	−0.141634	−0.0708170	+	0.997489i	$0.522561\pi$
$998$	0	0
$999$	4.47214	0.141492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4800.2.a.cv.1.2		2
4.3	odd	2		4800.2.a.cu.1.1		2
5.2	odd	4		960.2.f.k.769.2		4
5.3	odd	4		960.2.f.k.769.3		4
5.4	even	2		4800.2.a.cu.1.2		2
8.3	odd	2		2400.2.a.bj.1.2		2
8.5	even	2		2400.2.a.bi.1.1		2
15.2	even	4		2880.2.f.v.1729.1		4
15.8	even	4		2880.2.f.v.1729.4		4
20.3	even	4		960.2.f.k.769.1		4
20.7	even	4		960.2.f.k.769.4		4
20.19	odd	2	inner	4800.2.a.cv.1.1		2
24.5	odd	2		7200.2.a.cc.1.2		2
24.11	even	2		7200.2.a.cq.1.1		2
40.3	even	4		480.2.f.e.289.4	yes	4
40.13	odd	4		480.2.f.e.289.2	yes	4
40.19	odd	2		2400.2.a.bi.1.2		2
40.27	even	4		480.2.f.e.289.1	✓	4
40.29	even	2		2400.2.a.bj.1.1		2
40.37	odd	4		480.2.f.e.289.3	yes	4
60.23	odd	4		2880.2.f.v.1729.3		4
60.47	odd	4		2880.2.f.v.1729.2		4
80.3	even	4		3840.2.d.bh.2689.2		4
80.13	odd	4		3840.2.d.bg.2689.1		4
80.27	even	4		3840.2.d.bh.2689.1		4
80.37	odd	4		3840.2.d.bg.2689.2		4
80.43	even	4		3840.2.d.bg.2689.4		4
80.53	odd	4		3840.2.d.bh.2689.3		4
80.67	even	4		3840.2.d.bg.2689.3		4
80.77	odd	4		3840.2.d.bh.2689.4		4
120.29	odd	2		7200.2.a.cq.1.2		2
120.53	even	4		1440.2.f.h.289.2		4
120.59	even	2		7200.2.a.cc.1.1		2
120.77	even	4		1440.2.f.h.289.3		4
120.83	odd	4		1440.2.f.h.289.1		4
120.107	odd	4		1440.2.f.h.289.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.f.e.289.1	✓	4	40.27	even	4
480.2.f.e.289.2	yes	4	40.13	odd	4
480.2.f.e.289.3	yes	4	40.37	odd	4
480.2.f.e.289.4	yes	4	40.3	even	4
960.2.f.k.769.1		4	20.3	even	4
960.2.f.k.769.2		4	5.2	odd	4
960.2.f.k.769.3		4	5.3	odd	4
960.2.f.k.769.4		4	20.7	even	4
1440.2.f.h.289.1		4	120.83	odd	4
1440.2.f.h.289.2		4	120.53	even	4
1440.2.f.h.289.3		4	120.77	even	4
1440.2.f.h.289.4		4	120.107	odd	4
2400.2.a.bi.1.1		2	8.5	even	2
2400.2.a.bi.1.2		2	40.19	odd	2
2400.2.a.bj.1.1		2	40.29	even	2
2400.2.a.bj.1.2		2	8.3	odd	2
2880.2.f.v.1729.1		4	15.2	even	4
2880.2.f.v.1729.2		4	60.47	odd	4
2880.2.f.v.1729.3		4	60.23	odd	4
2880.2.f.v.1729.4		4	15.8	even	4
3840.2.d.bg.2689.1		4	80.13	odd	4
3840.2.d.bg.2689.2		4	80.37	odd	4
3840.2.d.bg.2689.3		4	80.67	even	4
3840.2.d.bg.2689.4		4	80.43	even	4
3840.2.d.bh.2689.1		4	80.27	even	4
3840.2.d.bh.2689.2		4	80.3	even	4
3840.2.d.bh.2689.3		4	80.53	odd	4
3840.2.d.bh.2689.4		4	80.77	odd	4
4800.2.a.cu.1.1		2	4.3	odd	2
4800.2.a.cu.1.2		2	5.4	even	2
4800.2.a.cv.1.1		2	20.19	odd	2	inner
4800.2.a.cv.1.2		2	1.1	even	1	trivial
7200.2.a.cc.1.1		2	120.59	even	2
7200.2.a.cc.1.2		2	24.5	odd	2
7200.2.a.cq.1.1		2	24.11	even	2
7200.2.a.cq.1.2		2	120.29	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.00000 q^{7} +1.00000 q^{9} +4.47214 q^{11} -4.47214 q^{13} -4.47214 q^{17} -2.00000 q^{21} -4.00000 q^{23} +1.00000 q^{27} -4.00000 q^{29} +8.94427 q^{31} +4.47214 q^{33} +4.47214 q^{37} -4.47214 q^{39} +10.0000 q^{41} -4.00000 q^{43} -8.00000 q^{47} -3.00000 q^{49} -4.47214 q^{51} +4.47214 q^{53} -13.4164 q^{59} -10.0000 q^{61} -2.00000 q^{63} -8.00000 q^{67} -4.00000 q^{69} -8.94427 q^{71} -8.94427 q^{73} -8.94427 q^{77} -8.94427 q^{79} +1.00000 q^{81} -4.00000 q^{83} -4.00000 q^{87} +6.00000 q^{89} +8.94427 q^{91} +8.94427 q^{93} +17.8885 q^{97} +4.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{7} + 2 q^{9} - 4 q^{21} - 8 q^{23} + 2 q^{27} - 8 q^{29} + 20 q^{41} - 8 q^{43} - 16 q^{47} - 6 q^{49} - 20 q^{61} - 4 q^{63} - 16 q^{67} - 8 q^{69} + 2 q^{81} - 8 q^{83} - 8 q^{87} + 12 q^{89}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^7 + 2 * q^9 - 4 * q^21 - 8 * q^23 + 2 * q^27 - 8 * q^29 + 20 * q^41 - 8 * q^43 - 16 * q^47 - 6 * q^49 - 20 * q^61 - 4 * q^63 - 16 * q^67 - 8 * q^69 + 2 * q^81 - 8 * q^83 - 8 * q^87 + 12 * q^89

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